Shape keys influence over time along mesh?
You probably want a series of non-relative shape key (look for a relative checkbox).  Then you would animate the Evaluation Time to iterate/interpolate through the various intermediate shapes of the inflating balloon.  (see https://www.blender.org/manual/fr/animation/shape_keys.html#workflow-for-absolute-shape-keys )

It is possible to accomplish the same thing using relative shape keys, but you have to animate each one, and your application sounds like a textbook example of non-relative shape keys.

You might also be able to accomplish something similar by using an armature and scaling each bone in sequence (which would be almost like using relative shape keys).